State of the Art of ANCF Elements Regarding Geometric Description, Interpolation Strategies, Definition of Elastic Forces, Validation and the Locking Phenomenon in Comparison with Proposed Beam Finite Elements

Abstract

The focus of the present article lies on new enhanced beam finite element formulations in the absolute nodal coordinate formulation (ANCF) and its embedding in the available formulations in the literature. The ANCF has been developed in the past for the modeling of large deformations in multibody dynamics problems. In contrast to the classical nonlinear beam finite elements in literature, the ANCF does not use rotational degrees of freedom, but slope vectors for the parameterization of the orientation of the cross section. This leads to several advantages compared to the classical formulations, e.g. ANCF elements do not necessarily suffer from singularities emerging from the parameterization of rotations. In the classical large rotation vector formulation, the mass matrix is not constant with respect to the generalized coordinates. In the case of ANCF elements, a constant mass matrix follows, which is advantageous in dynamic analysis. In the standard geometrically exact formulation, the parameterization of rotations leads to a nonlinear term containing quadratic velocities. The so-called quadratic velocity vector vanishes for ANCF elements which is advantageous in dynamic applications. In the present article, existing beam finite element formulations are analyzed and improved to derive formulations which are able to solve industrial applications with high performance, efficiency and accuracy. The interest lies especially on finite element formulations for multibody dynamics systems that are capable of large deformations in order to derive accurate solutions of nonlinear engineering and research problems. Existing shear deformable ANCF beam finite elements show an overly stiff behavior caused by the locking phenomenon. Existing locking problems are discussed in this article and avoided in the proposed elements in order to gain accurate solutions. The state of the art of ANCF elements in literature is reviewed including the basic description of the kinematics, interpolation strategies and definition of elastic forces, but also problems and known disadvantages arising in the existing elements, as e.g. the locking phenomenon. Regarding the description of the elastic forces, the present article shows the two standard approaches in literature as well as new enhanced formulations to avoid locking: a continuum mechanics based formulation for the elastic forces with elimination of Poisson and shear locking, and an extended hybrid structural mechanics based formulation for the elastic forces including a term for penalizing shear and cross section deformation. The definition of the element kinematics and interpolation strategies in existing elements are discussed and compared to the according descriptions of the proposed elements. A comparison of the solution of the proposed finite elements to analytical solutions in the literature and to the solution retrieved from commercial finite element software have shown high accuracy and high order of convergence. The speed of convergence is evaluated regarding different interpolation strategies and different formulations for the elastic forces. The investigations show that the proposed elements have high potential for simulation of geometrically nonlinear problems arising from real-life multibody applications and therefore are highly competitive with existing elements in commercial finite element codes. It has to be mentioned here that most of the studies on nonlinear elements based on the ANCF in literature use linear constitutive laws. Regarding many applications in which geometric and material nonlinearities arise, elastic material models are not sufficient to represent the real problem accurately. For this reason, an extension of the material model is necessary in order to fulfill the requirements of current but also of future materials arising in engineering or research. An overview of existing nonlinear material models in literature can be found in the “Appendix”.

Appendix: Nonlinear Constitutive Laws

It has to be mentioned that most of the studies on nonlinear elements based on the ANCF in literature use linear constitutive laws. Regarding many applications in which geometric and material nonlinearities arise, elastic material models are not sufficient to represent the real problem accurately. For this reason, an extension of the material model is necessary in order to fulfill the requirements of current but also of future materials arising in engineering or research. See, e.g., [11, 19] for the formulations of elasticity, plasticity and viscoelasticity in continuum mechanics. Furthermore, see [83] for a discourse on nonlinear elasticity and [129] for a nonlinear constitutive model suitable for rubber-like materials.

There exist theoretical, but also application-driven approaches for different inelastic material models in literature, see [51] for an overview of the state of the art of developments in ANCF, not only regarding the definition of the elastic forces and kinematics of ANCF elements, but also for recent developments on material models. As an example of application-driven research in the field of elasto-plasticity in multibody systems, the simulation of vehicle crash-tests has to be mentioned due to its importance in everyday life, see e.g. [82]. In case of multibody systems described in the FFRF, Ambrosio and Nikravesh [2] introduced elasto-plasticity by means of the finite element method. The standard fully-parameterized ANCF element [107, 139] allows for a straightforward implementation of general nonlinear constitutive models in the formulation. In the case that a flexible body undergoes large rotations, but only small deformations, there exist developments on a linear visco-elastic constitutive material model for ANCF elements, see e.g. [33] or [34] in which the stress tensor is split into deviatoric and volumetric components. More general nonlinear material models for large strain problems are discussed in [83]. A general nonlinear constitutive model under the assumption of hyper-elasticity and isotropy for beams undergoing large displacements is described in [69]. There, three hyper-elasticity material models for structural elements are presented, which are based on a Neo-Hookean constitutive law for compressible and incompressible materials and a Mooney-Rivlin constitutive law for incompressible materials. Poisson modes coupling stretch and bending to cross section deformation are captured and therefore the presented material models are suitable for a wide range of physics and engineering applications, e.g., for analyzing the behavior of rubber-like materials. The Yeoh material model is used for the dynamic large deformation problem of rubber-like material, e.g., in [65]. In case of nonlinear material behavior, fatigue analysis and especially in plasticity problems, the accurate handling of stress components is essential. Sugiyama and Shabana [123] have shown the application of plasticity combined with the ANCF in flexible multibody systems using a Lagrangian plasticity formulation describing large elastic but only small plastic deformation. For large plastic strain problems, Simo and Hughes [111] presented a plasticity formulation which splits the displacement gradient into elastic and plastic components. Elasto-plasticity and a formulation for plasticity is described e.g. in [47], in which incremental strains are used to update the stress components in a correct manner. Nikravesh et al. [82] introduced a plastic material model for rigid multibody systems using plastic hinges. Gerstmayr et al. [41] discussed plasticity and damage of vibrating structures under guided rigid-body motions. A detailed investigation of elasto-plasticity for beams is presented in [42] investigating an elasto-plastic pendulum with geometric stiffening, which serves as a widely used benchmark problem in robotics. In [43], the influence of plasticity upon the vibrations and rotations of this latter mentioned pendulum is discussed.

1.1 Modeling of Biomechanical Materials

A consistent integration of non-linearly elastic constitutive laws in order to apply the ANCF for the modeling of biological materials is of great interest as well and is getting more and more important in research. Since nonlinear constitutive models can not be implemented in beam models based on the classical theories easily, the investigation of such models for biomechanical applications is not straightforward. The standard fully-parameterized ANCF element [107, 139] allows for the modeling of general nonlinear material models used for biological materials and is, e.g., applied to model the deformation of ligaments of knee joints in cyclic motion in [137]. In the latter formulation, a combination of a biomaterial model, an underlying cross section deformation and the clamping conditions between ligaments and bones during a large bending of the knee joint are discussed. The formulations and results of different clamped end conditions for a knee joint, as the partially clamped joint and the fully clamped joint, are compared in [58]. While the fully clamped joint can not reproduce cross section deformation, the partially clamped joint allows for cross section deformation. In [74], two different knee joint models are investigated for studying ligament dynamics. While the first model involves a linear Hookean material law and does not capture cross section deformations, the second model involves a Neo-Hookean material law and is able to capture more general deformation modes, as Poisson modes and cross section deformation modes as well. Again in [74], the dynamic coupling between ligament tension and cross section deformation is analyzed in a numerical example involving a large displacement finite element formulation based on the standard fully-parameterized ANCF element [107, 139]. The constitutive models in [69] can be applied for rubber-like materials and are also suitable for analysis of biological tissues, ligaments or muscles. The presented models in [69] are investigated in an eigenfrequency test for a cantilevered as well as a simply supported beam comparing first two bending frequencies, torsional and axial frequencies for the different nonlinear constitutive models, standard Hookean, elastic line approach and the analytical values. Furthermore, the examples of a stiff pendulum and a very flexible pendulum show the accuracy of the models and a simple tension test of a simply supported beam shows that using a linear material model instead of a nonlinear constitutive model leads to singular configurations.

Regarding the modeling of surgeries or movements of robotics, the understanding, simulation and guidance of very flexible components is of great interest, e.g., in aerospace and manufacturing applications. The modeling of a flexible pipe conveying fluid using a planar ANCF beam element is described in [118]. The latter model is used to optimize the dynamic response of a flexible manipulator in [132]. The vibration control of a similar flexible manipulator using piezo-electric actuators is discussed in Nachbagauer et al. [80].